Constant mean curvature discs with bounded area
Authors:
Rafael López and Sebastián Montiel
Journal:
Proc. Amer. Math. Soc. 123 (1995), 1555-1558
MSC:
Primary 53A10
DOI:
https://doi.org/10.1090/S0002-9939-1995-1286001-0
MathSciNet review:
1286001
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: It has been long conjectured that the two spherical caps are then only discs in the Euclidean three-space ${\mathbb {R}^3}$ with non-zero constant mean curvature spanning a round circle. In this work, we prove that it is true when the area of such a disc is less than or equal to that of the big spherical cap.
- João Lucas Barbosa and Manfredo do Carmo, Stability of minimal surfaces and eigenvalues of the Laplacian, Math. Z. 173 (1980), no. 1, 13–28. MR 584346, DOI https://doi.org/10.1007/BF01215521
- Fabiano Brito and Ricardo Sa Earp, Geometric configurations of constant mean curvature surfaces with planar boundary, An. Acad. Brasil. Ciênc. 63 (1991), no. 1, 5–19. MR 1115489
- Yu. D. Burago and V. A. Zalgaller, Geometric inequalities, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 285, Springer-Verlag, Berlin, 1988. Translated from the Russian by A. B. Sosinskiĭ; Springer Series in Soviet Mathematics. MR 936419
- Ricardo Earp, Fabiano Brito, William H. Meeks III, and Harold Rosenberg, Structure theorems for constant mean curvature surfaces bounded by a planar curve, Indiana Univ. Math. J. 40 (1991), no. 1, 333–343. MR 1101235, DOI https://doi.org/10.1512/iumj.1991.40.40017
- Erhard Heinz, On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectifiable boundary, Arch. Rational Mech. Anal. 35 (1969), 249–252. MR 246237, DOI https://doi.org/10.1007/BF00248159
- Nicolaos Kapouleas, Complete constant mean curvature surfaces in Euclidean three-space, Ann. of Math. (2) 131 (1990), no. 2, 239–330. MR 1043269, DOI https://doi.org/10.2307/1971494
- Nicholas J. Korevaar, Rob Kusner, and Bruce Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), no. 2, 465–503. MR 1010168
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53A10
Retrieve articles in all journals with MSC: 53A10
Additional Information
Keywords:
Constant mean curvature,
disc,
isoperimetric inequality
Article copyright:
© Copyright 1995
American Mathematical Society
